reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem
  (superior_setsequence A1).n \+\ (superior_setsequence A2).n c= (
  superior_setsequence(A1 (\+\) A2)).n
proof
  (superior_setsequence A1).n \+\ (superior_setsequence A2).n = Union (A1
  ^\n) \+\ (superior_setsequence A2).n by Th2
    .= Union (A1 ^\n) \+\ Union (A2 ^\n) by Th2;
  then
  (superior_setsequence A1).n \+\ (superior_setsequence A2).n c= Union ((
  A1 ^\n) (\+\) (A2 ^\n)) by Th11;
  then
  (superior_setsequence A1).n \+\ (superior_setsequence A2).n c= Union ((
  A1 (\+\) A2) ^\n) by Th7;
  hence thesis by Th2;
end;
